3,899 research outputs found
'How to feel safe': international students study migration
A variety of institutional and representational mechanisms are used in the construction of 'international students' and other 'migrants' or 'ethnic minorities' as two distinctive social categories. As part of these construction processes, the individuals affiliated with each group are located in different positions within the matrix of social power relations: they are granted with differential abilities to exercise their right to freedom of movement, and play different roles in the process of knowledge production. This article will explore how these processes occur in a specific context, through an autoethnographic account of the experiences of the author as an international student at the University of Amsterdam. This account suggests a thematic and methodological alternative to the safe position that the academic training as prospective migration scholars offers to students
Lower bounds for adaptive linearity tests
Linearity tests are randomized algorithms which have oracle access to the
truth table of some function f, and are supposed to distinguish between linear
functions and functions which are far from linear. Linearity tests were first
introduced by (Blum, Luby and Rubenfeld, 1993), and were later used in the PCP
theorem, among other applications. The quality of a linearity test is described
by its correctness c - the probability it accepts linear functions, its
soundness s - the probability it accepts functions far from linear, and its
query complexity q - the number of queries it makes. Linearity tests were
studied in order to decrease the soundness of linearity tests, while keeping
the query complexity small (for one reason, to improve PCP constructions).
Samorodnitsky and Trevisan (Samorodnitsky and Trevisan 2000) constructed the
Complete Graph Test, and prove that no Hyper Graph Test can perform better than
the Complete Graph Test. Later in (Samorodnitsky and Trevisan 2006) they prove,
among other results, that no non-adaptive linearity test can perform better
than the Complete Graph Test. Their proof uses the algebraic machinery of the
Gowers Norm. A result by (Ben-Sasson, Harsha and Raskhodnikova 2005) allows to
generalize this lower bound also to adaptive linearity tests. We also prove the
same optimal lower bound for adaptive linearity test, but our proof technique
is arguably simpler and more direct than the one used in (Samorodnitsky and
Trevisan 2006). We also study, like (Samorodnitsky and Trevisan 2006), the
behavior of linearity tests on quadratic functions. However, instead of
analyzing the Gowers Norm of certain functions, we provide a more direct
combinatorial proof, studying the behavior of linearity tests on random
quadratic functions..
MDS matrices over small fields: A proof of the GM-MDS conjecture
An MDS matrix is a matrix whose minors all have full rank. A question arising
in coding theory is what zero patterns can MDS matrices have. There is a
natural combinatorial characterization (called the MDS condition) which is
necessary over any field, as well as sufficient over very large fields by a
probabilistic argument.
Dau et al. (ISIT 2014) conjectured that the MDS condition is sufficient over
small fields as well, where the construction of the matrix is algebraic instead
of probabilistic. This is known as the GM-MDS conjecture. Concretely, if a zero pattern satisfies the MDS condition, then they conjecture that
there exists an MDS matrix with this zero pattern over any field of size
. In recent years, this conjecture was proven in
several special cases. In this work, we resolve the conjecture
Correlation Testing for Affine Invariant Properties on in the High Error Regime
Recently there has been much interest in Gowers uniformity norms from the
perspective of theoretical computer science. This is mainly due to the fact
that these norms provide a method for testing whether the maximum correlation
of a function with polynomials of
degree at most is non-negligible, while making only a constant number
of queries to the function. This is an instance of {\em correlation testing}.
In this framework, a fixed test is applied to a function, and the acceptance
probability of the test is dependent on the correlation of the function from
the property. This is an analog of {\em proximity oblivious testing}, a notion
coined by Goldreich and Ron, in the high error regime. In this work, we study
general properties which are affine invariant and which are correlation
testable using a constant number of queries. We show that any such property (as
long as the field size is not too small) can in fact be tested by Gowers
uniformity tests, and hence having correlation with the property is equivalent
to having correlation with degree polynomials for some fixed . We stress
that our result holds also for non-linear properties which are affine
invariant. This completely classifies affine invariant properties which are
correlation testable. The proof is based on higher-order Fourier analysis.
Another ingredient is a nontrivial extension of a graph theoretical theorem of
Erd\"os, Lov\'asz and Spencer to the context of additive number theory.Comment: 43 pages. A preliminary version of this work appeared in STOC' 201
On the Beck-Fiala Conjecture for Random Set Systems
Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for
random sparse set systems. Concretely, these are set systems ,
where each element lies in randomly selected sets of ,
where is an integer parameter. We provide new bounds in two regimes of
parameters. We show that when the hereditary discrepancy of
is with high probability ; and when the hereditary discrepancy of is with high probability
. The first bound combines the Lov{\'a}sz Local Lemma with a new argument
based on partial matchings; the second follows from an analysis of the lattice
spanned by sparse vectors
Bias vs structure of polynomials in large fields, and applications in effective algebraic geometry and coding theory
Let be a polynomial of degree in variables over a finite field
. The polynomial is said to be unbiased if the distribution of
for a uniform input is close to the uniform
distribution over , and is called biased otherwise. The polynomial
is said to have low rank if it can be expressed as a composition of a few lower
degree polynomials. Green and Tao [Contrib. Discrete Math 2009] and Kaufman and
Lovett [FOCS 2008] showed that bias implies low rank for fixed degree
polynomials over fixed prime fields. This lies at the heart of many tools in
higher order Fourier analysis. In this work, we extend this result to all prime
fields (of size possibly growing with ). We also provide a generalization to
nonprime fields in the large characteristic case. However, we state all our
applications in the prime field setting for the sake of simplicity of
presentation.
As an immediate application, we obtain improved bounds for a suite of
problems in effective algebraic geometry, including Hilbert nullstellensatz,
radical membership and counting rational points in low degree varieties.
Using the above generalization to large fields as a starting point, we are
also able to settle the list decoding radius of fixed degree Reed-Muller codes
over growing fields. The case of fixed size fields was solved by Bhowmick and
Lovett [STOC 2015], which resolved a conjecture of Gopalan-Klivans-Zuckerman
[STOC 2008]. Here, we show that the list decoding radius is equal the minimum
distance of the code for all fixed degrees, even when the field size is
possibly growing with
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